Spectral non-concentration near the top for unimodular random graphs
Mikolaj Fraczyk, Ben Hayes, Madhu Sudan, Yufei Zhao
TL;DR
This work extends spectral non-concentration near the top of the spectrum from finite bounded-degree graphs to infinite unimodular random graphs, proving that the spectral measure cannot place an atom at the top in the infinite setting. The authors develop and combine finite-graph techniques (net removal, moment methods) with unimodular-specific tools (mass-transport, local selection, extended interlacing) to obtain quantitative bounds on $\mu_G[(1-\theta)x, x]$ in both finite and infinite contexts, with stronger decay in infinite regular expanders. A key outcome is a direct link between spectral non-concentration and random-walk return probabilities on Cayley graphs and non-amenable groups, including optimality statements via rapid-decay properties. The results have implications for spectral distributions and mixing behavior on infinite networks and Cayley graphs, and offer a framework for further extending spectral-concentration phenomena to broader random-graph models.
Abstract
In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the top of the spectrum. We extend this result to infinite unimodular random graphs. As a corollary, the spectral distribution of the adjacency operator cannot have an atom at the top. For an infinite regular expander, we deduce that the singularity of the spectral measure at the top satisfies $μ_G[(1-θ)ρ,ρ] \lesssim θ^c$ for some constant $c>0$, where $ρ$ is the spectral radius of the adjacency operator of the graph. This implies new general estimates on the return probabilities of random walks.